Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.14680

Composite boson theory of Hall crystals and their transitions to Wigner crystals

Julian May-Mann, Sayak Bhattacharjee, S. Raghu

Published Mar 16, 2026• 0 citations
arXivHTMLPDF

Abstract

We consider the crystallization of a two-dimensional electron system in a perpendicular magnetic field using composite boson theory. There are three possible states to consider: the Hall liquid, the Wigner crystal, and the Hall crystal (a state with both broken translation symmetry and a quantized Hall response). Within composite boson theory, these states map onto a superconductor, a Mott insulator, and a supersolid of composite bosons respectively. We show that when a $ν= 1$ Hall liquid has a sufficiently soft roton, there is a first order transition to a triangular lattice Hall crystal. If we continue to decrease the roton mass, there is a continuous transition from the Hall crystal to a Wigner crystal. {When the Hall crystal exhibits the integer quantum Hall effect,} this transition {is} described by a free Dirac fermion and, at the critical point, the coupling to the phonons of the crystal is irrelevant, {in the {renormalization group} sense}. We extend this analysis to fractional $ν= 1/m$ Hall liquids. There, due to kinetic frustration arising from flux attachment, honeycomb lattice Hall crystals are preferred over triangular ones at intermediate interaction strength.

DOI
10.48550/arxiv.2603.14680

Related papers

First-order quantum Hall to Wigner crystal phase transition on a triangular lattice: An infinite density matrix renormalization group study

Gleb Fedorovich, Clemens Kuhlenkamp, Ataç İmamoğlu +1 more

In this work we study a system of interacting fermions on a triangular lattice in the presence of an external magnetic field. We assume electrons are spin polarized and fix a density of one third, with one unit of magnetic flux per particle. The infinite density matrix renormalization group algorithm is used to compute the ground state of this generalized Fermi-Hubbard model. Increasing the strength of the nearest-neighbor repulsion, we find a first-order transition between an integer quantum Hall phase and a crystalline, generalized Wigner crystal state. The first-order nature of the phase transition is consistent with a Ginzburg-Landau argument. We expect our results to be relevant for moir & eacute; heterostructures of two-dimensional materials.

Physical review. B./Physical review. B•Jun 2, 2025•1 citation

First order Quantum Hall to Wigner crystal phase transition on a triangular lattice: an iDMRG study

Gleb Fedorovich, Clemens Kuhlenkamp, Ataç İmamoğlu +1 more

In this work we study a system of interacting fermions on a triangular lattice in the presence of an external magnetic field. We neglect spin and fix a density of one third, with one unit of magnetic flux per particle. The infinite density matrix renormalization group algorithm is used to compute the ground state of this generalized Fermi-Hubbard model. Increasing the strength of the nearest-neighbor repulsion, we find a first order transition between an Integer Quantum Hall phase and a crystalline, generalized Wigner crystal state. The first-order nature of the phase transition is consistent with a Ginzburg-Landau argument. We expect our results to be relevant for moiré heterostructures of two-dimensional materials.

arXiv:2411.03748•Nov 6, 2024•0 citations

Anyon Crystals and Hall Crystals in a Periodic Potential

S Bhattacharjee, Julian May-Mann, S. Raghu

We obtain integer and fractional quantum Hall crystals as ground states of a two-dimensional electron system subject to a strong perpendicular magnetic field and a periodic potential. For certain fractional states, we show that the Hall crystal can constitute an anyon crystal, with a periodic ordering of well-defined anyons. We find that the latter states can be stabilized at odd denominator Landau level filling fractions when Landau level mixing is sufficiently weak, and near half-filling of the underlying lattice. These phases are obtained from a mean-field analysis of an effective lattice model of bosons attached to an odd number of flux quanta, which transmutes their statistics to that of electrons. In boson coordinates, the Hall crystal is a supersolid: a superfluid with charge order. Under strong interactions, vortex-anti-vortex pairs spontaneously nucleate in the supersolid, realizing a crystalline state of anyons.

arXiv:2607.19466•Jul 21, 2026•0 citations

Laughlin-Jastrow-correlated Wigner crystal in a strong magnetic field

Hangmo Yi, H. A. Fertig

We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power $m$ of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.

arXiv:cond-mat/9803098•Physical review. B, Condensed matter•Aug 15, 1998•65 citations

Fractional quantum Hall effect and featureless Mott insulators

A. A. Burkov

We point out and explicitly demonstrate a close connection that exists between featureless Mott insulators and fractional quantum Hall liquids. Using magnetic Wannier states as the single-particle basis in the lowest Landau level (LLL), we demonstrate that the Hamiltonian of interacting bosons in the LLL maps onto a Hamiltonian of a featureless Mott insulator on triangular lattice, formed by the magnetic Wannier states. The Hamiltonian is remarkably simple and consists only of short-range repulsion and ring-exchange terms.

arXiv:0911.4094•Physical Review B•Mar 11, 2010•11 citations

Quantum Phase Transitions in Two-Dimensional Electron

Systems Alexander Shashkin, Sergey Kravchenko

Alexander Shashkin Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia Sergey Kravchenko Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A. Two-dimensional (2D) electron systems are realized when the electrons are free to move in a plane but their motion perpendicular to the plane is quantized in a confining potential well. Quantum phase transitions realized experimentally in such systems so far include metal-insulator transitions in perpendicular magnetic fields, metal-insulator transition in zero magnetic field, and a possible transition to a Wigner crystal. The first transition is governed by the externally controlled electron density or magnetic field, while the other two are governed by the electron density. At low electron densities in 2D systems, the strongly-interacting limit is reached because the kinetic energy is overwhelmed by the energy of electron-electron interactions. The interaction strength is characterized by the ratio between the Coulomb energy and the Fermi energy, r∗s = Eee/EF . Assuming that the effective electron mass is equal to the band mass, the interaction parameter r∗s in the single-valley case reduces to the Wigner-Seitz radius, rs = 1/(πns)1/2aB, and therefore increases as the electron density, ns, decreases (here aB is the Bohr radius in the semiconductor). Possible candidates for the ground state of the system include a Wigner crystal characterized by spatial and spin ordering [1], a ferromagnetic Fermi liquid with spontaneous spin ordering [2], a paramagnetic Fermi liquid [3], etc. In the strongly-interacting limit (rs 1), no analytical theory has been developed to date. According to numerical simulations [4], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. Refined numerical simulations [5] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ≤ rs ≤ 35, the ground state of the system is a strongly correlated ferromagnetic Fermi liquid. At higher electron densities, rs ∼ 1, the electron liquid is expected to be paramagnetic, with the effective mass, m, and Lande´ g factor renormalized by interactions. Apart from the ferromagnetic Fermi liquid, other intermediate phases between the Wigner crystal and the paramagnetic Fermi liquid may also exist.

CRC Press eBooks•Nov 2, 2010•1 citation
We consider the crystallization of a two-dimensional electron system in a perpendicular magnetic field using composite boson theory. There are three possible states to consider: the Hall liquid, the Wigner crystal, and the Hall crystal (a state with both broken translation symmetry and a quantized Hall response). Within composite boson theory, these states map onto a superconductor, a Mott insulator, and a supersolid of composite bosons respectively. We show that when a ν=1ν= 1ν=1 Hall liquid has a sufficiently soft roton, there is a first order transition to a triangular lattice Hall crystal. If we continue to decrease the roton mass, there is a continuous transition from the Hall crystal to a Wigner crystal. {When the Hall crystal exhibits the integer quantum Hall effect,} this transition {is} described by a free Dirac fermion and, at the critical point, the coupling to the phonons of the crystal is irrelevant, {in the {renormalization group} sense}. We extend this analysis to fractional ν=1/mν= 1/mν=1/m Hall liquids. There, due to kinetic frustration arising from flux attachment, honeycomb lattice Hall crystals are preferred over triangular ones at intermediate interaction strength.
In this work we study a system of interacting fermions on a triangular lattice in the presence of an external magnetic field. We assume electrons are spin polarized and fix a density of one third, with one unit of magnetic flux per particle. The infinite density matrix renormalization group algorithm is used to compute the ground state of this generalized Fermi-Hubbard model. Increasing the strength of the nearest-neighbor repulsion, we find a first-order transition between an integer quantum Hall phase and a crystalline, generalized Wigner crystal state. The first-order nature of the phase transition is consistent with a Ginzburg-Landau argument. We expect our results to be relevant for moir & eacute; heterostructures of two-dimensional materials.
In this work we study a system of interacting fermions on a triangular lattice in the presence of an external magnetic field. We neglect spin and fix a density of one third, with one unit of magnetic flux per particle. The infinite density matrix renormalization group algorithm is used to compute the ground state of this generalized Fermi-Hubbard model. Increasing the strength of the nearest-neighbor repulsion, we find a first order transition between an Integer Quantum Hall phase and a crystalline, generalized Wigner crystal state. The first-order nature of the phase transition is consistent with a Ginzburg-Landau argument. We expect our results to be relevant for moiré heterostructures of two-dimensional materials.
We obtain integer and fractional quantum Hall crystals as ground states of a two-dimensional electron system subject to a strong perpendicular magnetic field and a periodic potential. For certain fractional states, we show that the Hall crystal can constitute an anyon crystal, with a periodic ordering of well-defined anyons. We find that the latter states can be stabilized at odd denominator Landau level filling fractions when Landau level mixing is sufficiently weak, and near half-filling of the underlying lattice. These phases are obtained from a mean-field analysis of an effective lattice model of bosons attached to an odd number of flux quanta, which transmutes their statistics to that of electrons. In boson coordinates, the Hall crystal is a supersolid: a superfluid with charge order. Under strong interactions, vortex-anti-vortex pairs spontaneously nucleate in the supersolid, realizing a crystalline state of anyons.
We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power mmm of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.
We point out and explicitly demonstrate a close connection that exists between featureless Mott insulators and fractional quantum Hall liquids. Using magnetic Wannier states as the single-particle basis in the lowest Landau level (LLL), we demonstrate that the Hamiltonian of interacting bosons in the LLL maps onto a Hamiltonian of a featureless Mott insulator on triangular lattice, formed by the magnetic Wannier states. The Hamiltonian is remarkably simple and consists only of short-range repulsion and ring-exchange terms.
Alexander Shashkin Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia Sergey Kravchenko Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A. Two-dimensional (2D) electron systems are realized when the electrons are free to move in a plane but their motion perpendicular to the plane is quantized in a confining potential well. Quantum phase transitions realized experimentally in such systems so far include metal-insulator transitions in perpendicular magnetic fields, metal-insulator transition in zero magnetic field, and a possible transition to a Wigner crystal. The first transition is governed by the externally controlled electron density or magnetic field, while the other two are governed by the electron density. At low electron densities in 2D systems, the strongly-interacting limit is reached because the kinetic energy is overwhelmed by the energy of electron-electron interactions. The interaction strength is characterized by the ratio between the Coulomb energy and the Fermi energy, r∗s = Eee/EF . Assuming that the effective electron mass is equal to the band mass, the interaction parameter r∗s in the single-valley case reduces to the Wigner-Seitz radius, rs = 1/(πns)1/2aB, and therefore increases as the electron density, ns, decreases (here aB is the Bohr radius in the semiconductor). Possible candidates for the ground state of the system include a Wigner crystal characterized by spatial and spin ordering [1], a ferromagnetic Fermi liquid with spontaneous spin ordering [2], a paramagnetic Fermi liquid [3], etc. In the strongly-interacting limit (rs 1), no analytical theory has been developed to date. According to numerical simulations [4], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. Refined numerical simulations [5] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ≤ rs ≤ 35, the ground state of the system is a strongly correlated ferromagnetic Fermi liquid. At higher electron densities, rs ∼ 1, the electron liquid is expected to be paramagnetic, with the effective mass, m, and Lande´ g factor renormalized by interactions. Apart from the ferromagnetic Fermi liquid, other intermediate phases between the Wigner crystal and the paramagnetic Fermi liquid may also exist.